The answer seems to be positive according to this paper: Barnabás Farkas, Lajos Soukup: The zero density ideal, cardinal invariants and related forcing problems.¹

Theorem 2.3. If $\mathcal I$ is a rare ideal on $\mathbb N$ then $\mathfrak b = \mathfrak b_{\mathcal I}$ and $\mathfrak d = \mathfrak d_{\mathcal I}$.

Just before this theorem the authors mention mention that the ideal $\mathcal Z_0$ of the sets with zero density is a rare ideal.

A similar result is shown for analytic P-ideals in Corollary 5.5 of More on cardinal invariants of analytic P-ideals by the same two authors (arXiv, eudml). Again, this class of ideals includes $\mathcal Z_0$.

¹I wasn't able to find whether the paper was published somewhere, but a preprint is available here (Wayback Machine). The same paper was also mentioned in this answer: Are these two quotients of $\omega^\omega$ isomorphic?

The answer seems to be positive according to this paper: Barnabás Farkas, Lajos Soukup: The zero density ideal, cardinal invariants and related forcing problems.¹

Theorem 2.3. If $\mathcal I$ is a rare ideal on $\mathbb N$ then $\mathfrak b = \mathfrak b_{\mathcal I}$ and $\mathfrak d = \mathfrak d_{\mathcal I}$.

Just before this theorem the authors mention that the ideal $\mathcal Z_0$ of the sets with zero density is a rare ideal.

A similar result is shown for analytic P-ideals in Corollary 5.5 of More on cardinal invariants of analytic P-ideals by the same two authors (arXiv, eudml). Again, this class of ideals includes $\mathcal Z_0$.

¹I wasn't able to find whether the paper was published somewhere, but a preprint is available here (Wayback Machine). The same paper was also mentioned in this answer: Are these two quotients of $\omega^\omega$ isomorphic?

The answer seems to be positive according to this paper: Barnabás Farkas, Lajos Soukup: The zero density ideal, cardinal invariants and related forcing problems.¹

Theorem 2.3. If $\mathcal I$ is a rare ideal on $\mathbb N$ then $\mathfrak b = \mathfrak b_{\mathcal I}$ and $\mathfrak d = \mathfrak d_{\mathcal I}$.

Just before this theorem the authors mention mention that the ideal $\mathcal Z_0$ of the sets with zero density is a rare ideal.

¹I wasn't able to find whether the paper was published somewhere, but a preprint is available here (Wayback Machine). The same paper was also mentioned in this answer: Are these two quotients of $\omega^\omega$ isomorphic?

The answer seems to be positive according to this paper: Barnabás Farkas, Lajos Soukup: The zero density ideal, cardinal invariants and related forcing problems.¹

Theorem 2.3. If $\mathcal I$ is a rare ideal on $\mathbb N$ then $\mathfrak b = \mathfrak b_{\mathcal I}$ and $\mathfrak d = \mathfrak d_{\mathcal I}$.

Just before this theorem the authors mention mention that the ideal $\mathcal Z_0$ of the sets with zero density is a rare ideal.

A similar result is shown for analytic P-ideals in Corollary 5.5 of More on cardinal invariants of analytic P-ideals by the same two authors (arXiv, eudml). Again, this class of ideals includes $\mathcal Z_0$.

¹I wasn't able to find whether the paper was published somewhere, but a preprint is available here (Wayback Machine). The same paper was also mentioned in this answer: Are these two quotients of $\omega^\omega$ isomorphic?